Hardness of A/E-Design under Partition Constraints

2026-08-05

AI Summary

This study addresses the A/E-optimal experimental design problem under partition matroid constraints, which seeks a basis minimizing either the trace (A-design) or the largest eigenvalue (E-design) of the information matrix. By constructing a reduction from the 3-dimensional matching problem, the authors provide the first rigorous proof that this problem is inapproximable under partition constraints, thereby resolving an open question posed by Brown, Laddha, and Singh. This result highlights a fundamental computational distinction between A/E-optimal designs and D-optimal design, the latter admitting efficient approximation algorithms, and underscores the critical role of combinatorial structure in determining the tractability of experimental design problems.

Abstract

We consider the A/E-design problem under partition constraints: Given vectors $v_1,\ldots,v_N\in \R^d$ and a partition matroid on $[N]$, find a base $S$ of the matroid that minimizes $\tr(M(S)^{-1})$ or $λ_{\max}(M(S)^{-1})$ where $M(S)=\sum_{i \in S} v_i v_i^\top$. In contrast to D-design, where good estimation and approximation guarantees are known as a function of $d$, we show that no reasonable approximation exists for A/E-design. This answers a question of Brown, Laddha and Singh. The proof is based on an elementary reduction from three-dimensional matching.

Problem

• A/E-design
• partition constraints
• approximation hardness
• matroid optimization

Innovation

• A/E-design
• partition constraints
• hardness of approximation
• three-dimensional matching
• experimental design